Problem: Integrate. $ \int 4\csc^2(x)\,dx $ Choose 1 answer: Choose 1 answer: (Choice A) A $4\sec(x)+C$ (Choice B) B $4\tan(x)+C$ (Choice C) C $-4\cot(x)+C$ (Choice D) D $-4\csc(x)+C$
We need a function whose derivative is $4\csc^2(x)$. We know that the derivative of $\cot(x)$ is $-\csc^2(x)$, so let's start there: $\dfrac{d}{dx} \cot(x) = -\csc^2(x)$ Now let's multiply by $-4$ : $\dfrac{d}{dx} \left[ -4\cot(x) \right]= -4\dfrac{d}{dx} \cot(x) = 4\csc^2(x)$ Because finding the integral is the opposite of taking the derivative, this means that: $ \int 4\csc^2(x)\,dx =-4 \cot(x)\, + C$ The answer: $-4 \cot(x)\, + C$